Low-rank statistical finite elements for scalable model-data synthesis

Abstract

Statistical learning additions to physically derived mathematical models are gaining traction in the literature. A recent approach has been to augment the underlying physics of the governing equations with data driven Bayesian statistical methodology. Coined statFEM, the method acknowledges a priori model misspecification, by embedding stochastic forcing within the governing equations. Upon receipt of additional data, the posterior distribution of the discretised finite element solution is updated using classical Bayesian filtering techniques. The resultant posterior jointly quantifies uncertainty associated with the ubiquitous problem of model misspecification and the data intended to represent the true process of interest. Despite this appeal, computational scalability is a challenge to statFEM's application to high-dimensional problems typically experienced in physical and industrial contexts. This article overcomes this hurdle by embedding a low-rank approximation of the underlying dense covariance matrix, obtained from the leading order modes of the full-rank alternative. Demonstrated on a series of reaction-diffusion problems of increasing dimension, using experimental and simulated data, the method reconstructs the sparsely observed data-generating processes with minimal loss of information, in both the posterior mean and variance, paving the way for further integration of physical and probabilistic approaches to complex systems.

arXiv comment33 pages, 14 figures, revised version

References in corpus (11)

• SciPy 1.0--Fundamental Algorithms for Scientific Computing in Python
• Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
• Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations
• Learning data driven discretizations for partial differential equations
• Curse-of-dimensionality revisited: Collapse of the particle filter in very large scale systems
• Data Assimilation: A Mathematical Introduction
• Hilbert Space Methods for Reduced-Rank Gaussian Process Regression
• Evaluating Data Assimilation Algorithms
• The statistical finite element method (statFEM) for coherent synthesis of observation data and model predictions
• Kernel Operations on the GPU, with Autodiff, without Memory Overflows
• Parallel inversion of huge covariance matrices

Cited by in corpus (3)

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• Inferring Displacement Fields from Sparse Measurements Using the Statistical Finite Element Method
• Statistical reduced order modelling for the parametric Helmholtz equation